For the past several years, a rigorous couple wave approach (RCWA) and similar algorithms have been widely used for the study and design of diffraction structures. In the RCWA approach, the profiles of periodic structures are approximated by a given number of sufficiently thin planar grating slabs. Specifically, RCWA involves three main operations, namely, the Fourier expansion of the field inside the grating, calculation of the eigenvalues and eigenvectors of a constant coefficient matrix that characterizes the diffracted signal, and solution of a linear system deduced from the boundary matching conditions. RCWA divides the problem into three distinct spatial regions: (1) the ambient region supporting the incident plane wave field and a summation over all reflected diffracted orders, (2) the grating structure and underlying non-patterned layers in which the wave field is treated as a superposition of modes associated with each diffracted order, and (3) the substrate containing the transmitted wave field.
The accuracy of the RCWA solution depends, in part, on the number of terms retained in the space-harmonic expansion of the wave fields, with conservation of energy being satisfied in general. The number of terms retained is a function of the number of diffraction orders considered during the calculations. Efficient generation of a simulated diffraction signal for a given hypothetical profile involves selection of the optimal set of diffraction orders at each wavelength for both transverse-magnetic (TM) and/or transverse-electric (TE) components of the diffraction signal. Mathematically, the more diffraction orders selected, the more accurate the simulations. However, the higher the number of diffraction orders, the more computation is required for calculating the simulated diffraction signal. Moreover, the computation time is a nonlinear function of the number of orders used.
The input to the RCWA calculation is a profile or model of the periodic structure. In some cases cross-sectional electron micrographs are available (from, for example, a scanning electron microscope or a transmission electron microscope). When available, such images can be used to guide the construction of the model. However a wafer cannot be cross sectioned until all desired processing operations have been completed, which may take many days or weeks, depending on the number of subsequent processing operations. Even after all the desired processing operations are complete, the process to generate cross sectional images can take many hours to a few days because of the many operations involved in sample preparation and in finding the right location to image. Furthermore the cross section process is expensive because of the time, skilled labor and sophisticated equipment needed, and it destroys the wafer.
Thus, there is a need for a method for efficiently generating an accurate model of a periodic structure given limited information about that structure, a method for optimizing the parameterization of that structure and a method of optimizing the measurement of that structure.